Question

The Bears and the Hammers are playing a soccer game. At the end of regulation time the score is tied; this forces an over time. The rules for the overtime are straightforward. There can be at most 3 overtime periods. During the first overtime period each team is given one chance to score. If one team scores and the other does not score, the game is over and the scoring team wins the game. If this does not happen, then a second overtime period is played with each team again given one chance to score. As before if one team scores and the other does not score, the game is over and the scoring tea

Answer 1

The Bears and the Hammers are playing a soccer game. At the end of regulation time the score is tied; this forces an over time. The rules for the overtime are straightforward. There can be at most 3 overtime periods. During the first overtime period each team is given one chance to score. If one team scores and the other does not score, the game is over and the scoring team wins the game. If this does not happen, then a second overtime period is played with each team again given one chance to score. As before if one team scores and the other does not score, the game is over and the scoring team is the winner. If this does not happen, then a third overtime period is played with each team again given one chance to score. As before if one team scores and the other does not, the game is over and the scoring team wins the game. If this does not happen, then the game is over and is an official tie. The Bears are a better team than the Hammers. The Bears score 70% of the time in an overtime period and the Hammers score 65% of the time in an overtime period. Assume that the even of either team scoring in an overtime period is independent of the event that the other team scores in that overtime period.
a) Find Pr(the Hammers win the game at the end of the first overtime period)
b) Find Pr( Bears win the game)
c) Find Pr ( the game is an official tie )

Answer 2

so my logic was, first I made a contingency table

Answer 3

for a) i.e. hammers to win the game in 1st tiebreaker, requirement is that hammers score and bears dont,hence req probab = 0.65 * 0.3

Answer 4

thats what i got too , q

Answer 5

i had trouble with b)

Answer 6

b ) .459032

Answer 7

for b) i.e. bears win the game, they can win before overtime (info about that isn't given, so we neglect it) , they can win in 1st attempt, or 2nd or third
If they win in 1st attempt : 0.7 * 0.35
If they win in 2nd attempt : (0.3* 0.35 * 0.7 *0.35)
If they win in 3rd attempt : (0.3 * 0.35 * 0.3 * 0.35 * 0.7 *0.35)
Net probab = sum of these

Answer 8

Tell me this, is there a possibility that both score in a tie breaker and its a tie ?

Answer 9

Or is it simply that one who scores first wins ?

Answer 10

yes both can score , or both do not score , then its tie (go to next overtime)

Answer 11

I see, I missed that case, leme try again

Answer 12

no, it goes if one team scores AND the other team does not score, then its a win.

Answer 13

here is my cases
P( bears win 1st O.T or 1st O.T is tie and bears win 2nd O.T. or 1st & 2nd is tie and bears win 3rd O.T. ) , these are mutually exclusive

Answer 14

OT = overtime

Answer 15

for b) i.e. bears win the game, they can win before overtime (info about that isn't given, so we neglect it) , they can win in 1st attempt, or 2nd or third
If they win in 1st attempt : 0.7 * 0.35
If they win in 2nd attempt : ((0.3* 0.35)+(0.7*0.65)) * 0.7 *0.35)
If they win in 3rd attempt : ((0.3*0.35) + (0.7 *0.65)) * ((0.3*0.35) + (0.7 *0.65)) * (0.7 *0.35)

net = sum of these = 0.459032

Answer 16

Yep your logic is correct.

Answer 17

C) is pretty straightforward . So, did you follow sire ?

Answer 18

one second, let me look over your work

Answer 19

how did you get this, please explain
If they win in 2nd attempt : ((0.3* 0.35)+(0.7*0.65)) * 0.7 *0.35)

Answer 20

oh i see, thats your tie

Answer 21

yep

Answer 22

so bears winning second overtime depends on getting tie on first overtime, but i thought they were independent

Answer 23

Well of course not, for any other result other than tie in 1st OT , we won't proceed to 2nd OT.

Answer 24

why isnt the second one
P ( bears win 2nd overtime | first overtime is tie )

Answer 25

Thats is what we have done, isn't it ?

Answer 26

not quite

Answer 27

we did
P ( first overtime is a tie & bears win 2nd overtime )

Answer 28

Yes, whats the difference ?

Answer 29

P( bears win 1st or 1st is tie and bears win 2nd or 1st & 2nd is tie and bears win 3rd ) =
these are mutually exclusive

Answer 30

P( bears win 1st OR 1st is tie and bears win 2nd OR 1st & 2nd is tie and bears win 3rd )= P ( bears win 1st ) + P ( bears win 2nd and 1st is tie) + P ( bears win 3rd and 1st & second is tie )

Answer 31

See, frankly speaking, I don';t understand much of exclusive and " | " notations, I am not understanding your query,

Answer 32

bears can only win 2nd if 1st was a tie

Answer 33

so tie comes first

Answer 34

(0.35 *0.3) + (0.65 * 0.7)